# What are the consequences of knowing (or not knowing) the range of possible answers from a test

Lets say we are interested in some unknown variable x. We poll a x at different intervals and get a set of values that x has been. However we know that if we poll x enough times we will eventually get repetition since x is random. What can be gained from knowing the upper bound of x.

A brief example: Lets say I want to take a snapshot of the magnitude of the acceleration of a car. I can do this at any time. I might take 10, 50 or 1000 snapshots. None of these can defiantly tell me the upper bound of the acceleration, although the more samples I take the more confident I can be that I've either captured it, or am near it. Now, lets say I know the local geography, specifically the steepest hills in the area. I also have the cars schematic. With these two pieces of information I can (in theory) calculate the maximum acceleration of the car. Is this beneficial in any way to my statistics? The lower bound is clearly 0, that is known before any samples or other information. What new data can I learn from knowing the upper bound?

Originally my intuition was that the upper bound would help me gain details of the population size, but they now seem completely unrelated.

More detail on my problem:

Lets assume I want to predict the number of car crashes on a given stretch of road. If I say there's a small chance of any 2 cars colliding, based on the make of the car, the relative speed etc. then I could simulate every possible eventuality. For example, car 'A' collides with one car in a set of simulations, but the car it collides with ranges through the population of cars. There'll also be the case where car 'A' collides with car 'B' and in the same simulation car 'C' collides with car 'D' (ie. 2 collisions in 1 simulation) this may be less likely but still possible. Now given the total population of cars, and the number of time steps in the simulation I could work out how many possible different results there are. Some results will occur more than once in a random collection of simulations, but in terms of unique simulations there should be a fixed, finite number. But if I can calculate this number what, if anything, does it teach me about the problem?

• Should read 'set of values'. Edited! – FraserOfSmeg Apr 7 '14 at 18:48
• "Interested in" and "what can be gained" are rather vague. Could you be more precise about what you want to learn about $x$? – whuber Apr 7 '14 at 18:51
• @whuber I can't really be more precise. I think I've found a method to calculate the number of possible answers/results. But this would be time consuming to implement and I'm curious to see if it's beneficial before I spend the time on it. Statistics isn't my field at all, so I'm unsure of the possible benefits. – FraserOfSmeg Apr 7 '14 at 20:32
• The problem is that without being precise or specific there isn't much of a question here. It's easy to imagine different kinds of variables $x$ that would require quite different types of answers and it's impossible in any event to determine what can be "gained" if you are not free to share your objectives with us. – whuber Apr 7 '14 at 20:33
• @whuber I didn't want to make the problem so specific that it's of no use to anyone else. I've added more detail now though. Thanks! – FraserOfSmeg Apr 7 '14 at 23:54

What you are thinking of is the "playground" or "spielraum". It does not help you "statistically" but it helps you "logically" in determining how much corroboration/verisimilitude your theory deserves if the data matches predictions. The vaguer the prediction (the more space of the spielraum it is consistent with) the less impressive it is that it fits with the data. See figures three and four from this paper:

Paul Meehl. 1990. Appraising and Amending Theories: The Strategy of Lakatosian Defense and Two Principles That Warrant It. Psychological Inquiry 1990, Vol. 1, No. 2, 108-141

• Thank you for this very complete answer. It's really great to get an answer that includes a reference for further reading. Having looked over the reference I feel I may have a few follow-up questions. Once I've gone over it all I'll post in the comments if you don't mind. Thank again! – FraserOfSmeg Apr 8 '14 at 15:37
• @FraserOfSmeg No problem. Discovering Paul Meehl clarified my thoughts on many matters. You should keep in mind the context though. AKAIK his concept of the spielraum is not universally accepted because it involves the "principle of indifference". However, it seems to me that the alternative suggested is to ignore the "logical" aspects of the inference problem exist at all. This has lead to much confusion amongst researchers. Meehl seems to be on the right track. – Livid Apr 9 '14 at 4:41
• Ok my first sub-question: I've gone of the maths in figure 3. The numbers all work out fine, but I can't find the source of the 0.44 and 0.56 used for the normalised collaboration index. Where do these two numbers come from? Second sub-question: if I run 100 tests to produce my theory (lets say it's value M+- 10) then why value do I use for the observation. I could run a further 100 tests and see if any of these extend out of my theory, but I can't see the benefit of this - is there one? – FraserOfSmeg Apr 9 '14 at 15:38
• @FraserOfSmeg Meehl shows the calculation on page 129 (equations 1-3). That is for a directional hypothesis (1/2 the speilraum consistent with the theory) while his example in figure 3 uses a more precise prediction. I think his calculation there is incorrect. Personally I have not really used that index, imo the reliance on the principle of indifference assumption limits the utility of any type of calculation, getting some exact value under these circumstances even seems kind of misleading. – Livid Apr 9 '14 at 22:34
• @FraserOfSmeg I think of it in more qualitative terms. Say your theory/model predicts 5-8 accidents and you observe 7. If there can be a max of 10 accidents, this is less impressive than if there can be a max of 1000 accidents. Similarly it would be less impressive if a theory predicting 3-10 accidents was consistent with the data. – Livid Apr 9 '14 at 22:38